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 Mat. Zametki, 1977, Volume 22, Issue 2, Pages 231–244 (Mi mz8044)

Uniform regularization of the problem of calculating the values of an operator

V. V. Arestov

Institute of Mathematics and Mechanics, Ural Scientific Center of the AS of USSR

Abstract: Let $X$ and $Y$ be linear normed spaces, $W$ a set in $X$, $A$ an operator from $W$ into $Y$, and $\mathfrak M$ the set $\mathfrak G$ of all operators or the set $\mathscr L$ of linear operators from $X$ into $Y$. With $\delta\ge0$ we put
$$\nu(\delta,\mathfrak M)=\inf_{T\in\mathfrak M}\sup_{x\in W}\sup_{\|\eta-x\|_X\le\delta}\|Ax-T\eta\|_Y.$$
We discuss the connection of $\nu(\delta,\mathfrak M)$ with the Stechkin problem on best approximation of the operator $A$ in $W$ by linear bounded operators. Estimates are obtained for $\nu(\delta,\mathfrak M)$ e.g., we write the inequality, where $H(Y)$ is Jung's constant of the space $Y$, and $\Omega(t)$ is the modulus of continuity of $A$ in $W$.

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English version:
Mathematical Notes, 1977, 22:2, 618–626

Bibliographic databases:

UDC: 517.5

Citation: V. V. Arestov, “Uniform regularization of the problem of calculating the values of an operator”, Mat. Zametki, 22:2 (1977), 231–244; Math. Notes, 22:2 (1977), 618–626

Citation in format AMSBIB
\Bibitem{Are77} \by V.~V.~Arestov \paper Uniform regularization of the problem of calculating the values of an operator \jour Mat. Zametki \yr 1977 \vol 22 \issue 2 \pages 231--244 \mathnet{http://mi.mathnet.ru/mz8044} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=493442} \zmath{https://zbmath.org/?q=an:0357.47017} \transl \jour Math. Notes \yr 1977 \vol 22 \issue 2 \pages 618--626 \crossref{https://doi.org/10.1007/BF01780971} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. V. V. Arestov, “Approximation of unbounded operators by bounded operators and related extremal problems”, Russian Math. Surveys, 51:6 (1996), 1093–1126
2. O. A. Timoshin, “The best approximation to the operator of the second mixed derivative”, Izv. Math., 62:1 (1998), 191–200
3. G. V. Khromova, “On the moduli of continuity of unbounded operators”, Russian Math. (Iz. VUZ), 50:9 (2006), 67–74
4. Babenko Yu., Skorokhodov D., “Stechkin's Problem for Differential Operators and Functionals of First and Second Orders”, J. Approx. Theory, 167 (2013), 173–200
5. V. I. Maksimov, “Calculation of the derivative of an inaccurately defined function by means of feedback laws”, Proc. Steklov Inst. Math., 291 (2015), 219–231
6. R. R. Akopian, “Optimal recovery of an analytic function in a doubly connected domain from its approximately given boundary values”, Proc. Steklov Inst. Math. (Suppl.), 296, suppl. 1 (2017), 13–18
7. R. R. Akopian, “Optimal Recovery of Analytic Functions from Boundary Conditions Specified with Error”, Math. Notes, 99:2 (2016), 177–182
8. R. R. Akopyan, “Optimal recovery of a function analytic in a disk from approximately given values on a part of the boundary”, Proc. Steklov Inst. Math. (Suppl.), 300, suppl. 1 (2018), 25–37
9. Akopyan R.R., “Optimal Recovery of a Derivative of An Analytic Function From Values of the Function Given With An Error on a Part of the Boundary”, Anal. Math., 44:1 (2018), 3–19
10. R. R. Akopyan, “Optimalnoe vosstanovlenie analiticheskoi v poluploskosti funktsii po priblizhenno zadannym znacheniyam na chasti granichnoi pryamoi”, Tr. IMM UrO RAN, 24, no. 4, 2018, 19–33
11. R. R. Akopyan, “An analogue of the two-constants theorem and optimal recovery of analytic functions”, Sb. Math., 210:10 (2019), 1348–1360
12. V. V. Arestov, R. R. Akopyan, “Zadacha Stechkina o nailuchshem priblizhenii neogranichennogo operatora ogranichennymi i rodstvennye ei zadachi”, Tr. IMM UrO RAN, 26, no. 4, 2020, 7–31
13. R. R. Akopyan, “Analog teoremy Adamara i svyazannye ekstremalnye zadachi na klasse analiticheskikh funktsii”, Tr. IMM UrO RAN, 26, no. 4, 2020, 32–47
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